## Tuesday, October 19, 2004 ... //

### MOND and holography

These blogs should be sources of provocative ideas that can lead to something new, which is why this article is about MOND and holography. Be sure that I find all MOND theories very unlikely but we should all be aware of interesting speculations that have shown non-trivially looking quantitative results.

Note added in 2006: more direct observations of dark matter have arguably falsified all published versions of MOND, great, and it is a somewhat open question whether more sophisticated versions of MOND like mine - that may involve macroscopic interference - are gone, too. The question can be settled by similar observations of the clusters in the future: if the distribution of the deduced dark matter is going to be random and independent of the visible matter, then any theory without dark matter will be dead.

What is MOND? The acronym MOND stands for "modified Newtonian dynamics", and it is an alternative to the theory of dark matter. Jacob Bekenstein gave a talk about MOND at Harvard when he was visiting us - and he is probably the most famous current advocate of this theory which is viewed as a controversial one: MOND does not quite agree with the usual picture of the Universe that emerges from general relativity. There is a lot of reasons to think that MOND is nonsense, but the goal of this article is just the opposite one. ;-)

Basic argument in favor of dark matter

In order to convince you that there can be something nice about MOND, you need to know the basic reasons why most people believe that there is a lot of dark matter around.

When we observe other galaxies, we can use the usual laws of general relativity (well, Newton's laws are enough) to predict the angular velocity of the stars in that particular galaxy as a function of their distance from the center. We think that we know the mass distribution because we know where the stars are located and what their mass is, and therefore we may calculate the gravitational field, and the motion of the stars in this gravitational field.

What results do we get from Newton's theory if we substitute the masses of the visible stars as the source of the gravitational field? The result says that the angular velocity (the velocity divided by the radius) decreases with the distance from the center (with the radius), much like in the case of planets orbiting the Sun. You know that Pluto's motion around the Sun is much slower than the motion of Mercury, and the ratio of the angular velocities is even more pronounced because Pluto is much further from the Sun and the distance appears in the denominator.

However, the direct observations of the stars' motion lead to a different result: the angular velocity is nearly constant for all stars in a single galaxy, except for a few stars near the galactic center. The rotation of a galaxy resembles the rotation of an LP recording! Although the gravitational laws have been successfully tested for planets and many other celestial bodies, the predictions of the rotation of galaxies seem to fail.

The usual response is that the true gravitational field is very different from the gravitational field of the visible stars: there is a huge amount of invisible matter (dark matter) whose gravitational field is such that the stars in the galaxy rotate like points on an LP recording. One must work with a reasonable distribution of dark matter to achieve this goal, and the theories of dark matter are pretty messy. Most of the dark matter must occupy a "halo" that is pretty far from the galactic center, and one needs to tune many parameters to make the dark matter theory agree with all the observed galaxies. Well, that's what you're forced to do if you believe that general relativity is a good description at very long, cosmological distances.

MOND basic achievements

One can measure the dependence of angular velocity on the distance from the center for different galaxies. Each galaxy gives a slightly different result. The predictions of Newton's theory are pretty good for the stars near the center - but a totally different behavior takes over once the distance from the center is greater than a certain critical distance.

In fact, the point where the angular velocity starts to deviate from Newton's laws - where it starts to be constant - always seems to be at the distance where the acceleration of the stars decreases to a certain universal critical acceleration a_0. For each galaxy, the distance of the first stars for which Newton's laws "break down" is slightly different; but their acceleration is always the same, with a rather good accuracy. That's interesting, is not it?

Moreover, the constant a_0 is a number pretty close to the Hubble constant H. In other words, a_0 is comparable to the inverse age of the Universe (multiplied by the speed of light). This fact will be important at the end of this posting.

The MOND theories, roughly speaking, say that the Newtonian laws are modified in such a way that the inverse square law (1/r^2) for gravity is continously replaced by the inverse distance law (1/r) for all objects whose acceleration is smaller than the critical value a_0. This crazy assumption allows one to reproduce the observed results from a large number of galaxies, using a minimal number of parameters - and without assuming any dark matter. (The agreement is not so great for clusters of galaxies, but this is not what we want to analyze here.)

Many ways how to derive this weird new behavior from "more field-theoretical" laws, closer to GR, have been proposed - but let me say that neither of them looks natural enough. Therefore the phenomenological description with the 1/r^2 force transmuting into 1/r for small accelerations is everything we find valuable about MOND.

Why it may follow from holography?

OK, let me now switch to my speculative explanation why this weird behavior may be derivable from holography. (Jacob Bekenstein told me that a crackpot has already written a paper about this relation MOND-holography, and therefore I apologize if I am not the first one who proposes this idea haha.)

First, imagine the usual holograms in optics, invented by Dennis Gabor in the 1950s. They are two-dimensional pictures. But if you look at them carefully (in some cases, you will need laser - the same laser that is necessary to produce them), you see a three-dimensional object.

The three-dimensional illusion results from a dense network of interference patterns. These patterns reflect that wave character of light. A small piece of the hologram can still be used to reconstruct the whole three-dimensional image, although the quality is reduced. It will be important below to keep in mind that if the piece of the hologram is way too small (compared to the object that we want to see, or even compared to the wavelength of the light), the three-dimensional illusion will disappear.

In quantum gravity, we know that something like holography is important, too. 't Hooft and Susskind were the first to propose the idea (the holographic principle) that in every theory of quantum gravity (which really means in "every solution of string/M-theory"), the information about the "bulk" can be encoded on the surface of this volume, and the density of information is never bigger than roughly one bit per Planck area.

Normally, we think that the "memory" grows with the volume. The bigger volume we have, the more RAM chips we can insert into this volume. However, gravity guarantees that this can't work indefinitely. If you put too many RAM chips into a too small volume, the gravity will be so strong that the chips will collapse into a black hole. It has been realized by Bekenstein and Hawking that the entropy (or information) carried by this black hole only scales like r^2 (in four spacetime dimensions) divided by the Planck area, instead of the usual r^3 behavior. Black hole is, at the same moment, the most entropic system that can fit a given volume. It really means that large volumes can carry much less information than what we would expect from a naive proportionality law (with the volume).

Maldacena's AdS/CFT correspondence is a very rigorous example of holography in string theory: in its most popular version, a four-dimensional (gauge) theory living on the boundary of five-dimensional anti de Sitter space contains all information about quantum gravitational (and stringy) physics inside the five-dimensional "bulk" (multiplied by another five-dimensional sphere, to get the total of ten dimensions).

Forget about AdS/CFT for a while, and think about our real Universe again. OK, I'm now gonna argue that MOND may follow from holography. As I mentioned above, the three-dimensional illusion of a hologram in optics breaks down if the piece of your hologram is too small. It's not unnatural to believe that a similar limitation occurs for holograms in quantum gravity. More generally, I want to argue that holography in quantum gravity implies modifications of dynamics in the "infrared" - and we want to define "infrared" according to the acceleration.

Consider an object whose acceleration is a. The worldline in spacetime is a hyperbola, and the center of curvature of this hyperbola is at distance 1/a. If you associate de Broglie's wave with that object, the lines of constant phase will depend on the velocity, and they will intersect at the center of curvature of that hyperbola. Let's now believe that this self-intersection is necessary for the three-dimensional interpretation of our hologram to be valid. OK?

If you swallowed that, then we're done. It's simply because the 3+1-dimensional physics can only be trusted if the center of the hyperbola fits into the cosmic hologram, i.e. if 1/a is smaller than the radius of the Universe. In other words, the usual physics only occurs if a is greater than the critical acceleration. If the acceleration is smaller, you're not allowed to use the 3+1-dimensional laws to calculate the forces affecting the object. Instead, you should switch to the physics of the hologram which is 2+1-dimensional. If you allow me to make one more leap, I can even say that the usual 1/r^2 force in 3+1 dimensions is replaced by the 1/r force in 2+1 dimensions (of the hologram), which is precisely what we need for MOND to describe the rotation of galaxies without any dark matter.

#### snail feedback (7) :

Dear Lubos,

I collected the following basic points.

a) Constant velocities for stars in galactic halo appear
for some critical acceleration a_0.

b) You consider an explanation for this transition in
terms of effective 2-dimensionality. As if the stars were moving in 2-D space instead of 3-D space and obeying 1/r law.

As an inhabitant of a Universe described by Topological
Geometrodynamics, I resonate with effective 2-dimensionality since all data about states of quantum
states in this particular Universe can be coded in terms of 2-surfaces: they can represent partons, boundaries of macroscopic objects, black holes, even boundary of large voids. This explains why information content of system seems to grow like area rather than volume.

The construction of S-matrix however requires 3-D
light-like surfaces representing the orbits of these
2-surfaces but only the normal derivatives of appropriate dynamical variables are needed. This would be mechanics for 2-D objects whose orbits are 3-D light-like surfaces. These 3-D surfaces defined boundary data defining 4-D space-time surface associated with them. Much like the frame codes the shape of the soap film.

Alternative explanations for v_0=constant law

I can imagine alternative explanations for the
v_0=constant law.

a) Dark matter consists of decay remnants of string like objects, whose net gravitational mass inside a sphere of radius R behaves as M(R) propto R, gives the same result. One could imagine that the original situation was a kind of mildly tangled spaghetti inside the sphere and the cosmic strings decayed to ordinary matter and magnetic flux tubes later. I have proposed this kind of explanation based on TGD counterparts of cosmic strings with string tension around 10^-7/G. See "Cosmic Strings" at
http://www.physics.helsinki.fi/~matpitka/tgd.html#cstrings.

b) An alternative model would involve a single very
massive more or less straight string like object orthogonal to the plane of galaxy creating a Newtonian gravitational field which would go like 1/r. GRT would predict only a lens effect.

Astral Bohr rules

How to explain why a_0 is quantized remains however a
challenge. The hint comes from experimental claims for
Bohr rule like quantization for the orbits of planets
around Earth. Bohr quantization might also explain the
universal value of a_0 using Bohr rules applied to starts in the galactic halo.

Nottale, the developer of scale relativity has proposed his own fractal model based on fractal hydrodynamics, see D. Da Roacha and L. Nottale (2003), Gravitational Structure Formation in Scale Relativity, astro-ph/0310036.

The key point is that hbar in Bohr rules is replaced with a gigantic value of hbar:

hbar_gr= GMm/v_0, v_0/c= 4.6*10^{-4} (I use c=1 in the following).

The general form of hbar_gr is forced by Equivalence
Principle. Also harmonics and sub-harmonics of v_0 appear. A possible interpretation is that the unit of angular momentum becomes gigantic. Equivalence Principle favors interpretation in terms of quantization of circulation using universal unit GM/v_0 determined by the large mass.

Is quantal dark matter behind astral Bohr rules?

I have developed my own model based on the hypothesis that a genuine quantum system is in question. See

"Gravitational Schroedinger Equation as a Quantum Model for the Formation of Astrophysical Structures and Dark
Matter?" at
http:/www.physics.helsinki.fi/~matpitka/nottale.pdf .

The motivation was the earlier number theory inspired
speculation that hbar could be dynamical, see

"Equivalence of Loop Diagrams with Tree Diagrams and
Cancellation of Infinities in Quantum TGD" at
http://www.physics.helsinki.fi/~matpitka/tgd.html#bialgebra.

1/hbar was conjectured to have values expressible in terms of Beraha numbers B(n)= 4*cos^2(pi/n), n=3,4,5... B(3) corresponds to 1/hbar=0. These numbers appear in context of quantum groups and von Neumann algebras.

The idea would be that the friendly Mother Nature takes
care of the problems caused by non-perturbative behavior and worrying theoreticians. As the interaction becomes too strong, a transition to a phase in which hbar is larger occurs (n-->n-1). At the last step n=4-->3 a transition to 1/hbar=about zero occurs. There is however a small perturbative correction to 1/hbar giving

1/hbar_gr= v_0/GMm

in the case of gravitation. Note that Mm must be above
Planck mass squared. This picture might apply also to
color confinement.

If this picture makes sense, the road to quantum gravity
might be the same modest bottom-up path leading to QED:
first Bohr rules, then Schroedinger equation, etc.. The
infrared problem for hydrogen atom would be replaced with the infrared problem for black holes: how Nature prevents the collapse of matter inside blackhole. It is indeed known that the models of super-novae tend to predict collapse to a black hole but this does not happen.

The matter in this non-perturbative phase would be dark
matter.

a) It would be quantum coherent since the time scales of quantum coherence are multiplied by a gigantic number
hbar_gr/hbar.

b) Darkness would follow from the extremely small value of also fine structure constant e^2/hbar_gr in this phase. Of course, it would be entire Bose-Einstein condensate that would couple with this coupling to em interactions.

c) The Bohr rules for the visible matter would reflect the astro-quantal behavior of the dark matter at larger
space-time sheets behaving like quantum rigid bodies having shape of a tube surrounding the orbit of planet say, and a detailed model for the evolution of planetary system follows. Quite strong predictions for follow for the inclinations and eccentricities of planets and comets just using semiclassical view about angular momentum.

d) The value of v_0 is essentially the ratio of Planck
length and CP_2 size in TGD Universe, and also its
harmonics and sub-harmonics find an explanation.

Universality of a_0 from Bohr rules

Returning now to the question about quantization of a_0.
The quantization would follow from the quantization of
radii and therefore of also acceleration of stellar orbits in the galactic halo: the acceleration would be

a_0 =v_0^2/r_0 ,

where r_0 is the Bohr radius for the smallest orbit for
which M(R) propto R for the density of matter is still
satisfied.

With Best Regards,

Matti Pitkanen

I think Hologrpahy as explained by Hooft, as shadows on the wall, is a interesting idea in relation to Plato's Cave.:)

One expands(looks out of the cave) and the defintion to remain consistent with the spactime understanding and of course, reduces all back to "include" all the dimensional references we like to think of, in those same unseen compactified dimensions?

We then have a interesting amount of dimensional information about the bulk, from the fifth dimensional value of that same light(we know gravity and light have been joined)?:)

Plato

The wastebook of A. Rivero ;-) contains a sheet which could be related to this theme. For two bodies revolving at a constant distance R one from another, he writes a sort of classical Newton or Kepler Law in natural units h=c=1

R^(5-D) Lp^(D-2) = LM Lm^2 Jm^2

where D=n+2 comes from an atractive force law G 1/R^n,
Lp is the natural lenght got from G via dimensional analysis,
Jm is the angular momentum of the system (measured from the body m?),
and LM, Lm are the Compton lengths of the bodies, or their mass if you prefer.

Now the wastebook notes that if Lm or LM are asked to be proportional to the lenght Lp, two different effects happen. If Lm= p Lp then
R^(5-D) Lp^(D-4) = LM p^2 Jm^2
while if LM=q Lp we have
R^(5-D) Lp^(D-3)= q Lm^2 Jm^2

My new remark in reviewing this sheet is: please note that in the former equation, the length Lp cancels when D=4, ie for 1/R^2 force. And in the later, Lp cancels when D=3, ie for 1/R force. So to mantain some independence of Lp (or, if you prefer, of G) we must to change from one force to another depending of which body is in some sense more relevant (depending of distance, perhaps?).

Of course if we cuantise both lengths then the equation is unuseful,
R^(5-D) Lp^(D-5) = q p^2 Jm^2
because R disappears also, so it is not possible to infer a relationship between angular momentum and radius of the orbit

footnote: the notebook also tells that M/m=Lm/LM=(JM/Jm)^2, if we want to change the system of reference to use the angular momentum JM instead of Jm. I guess this was all trivial, but I am just copying, not reproducing the reasonment :-)

Second footnote: let me to insist that Lm and LM refer to the quantities h/mc and h/Mc respectively, and they are not related, at least directly, to the owner of the Blog :-)))) nor to any clone of him.